This paper considers the tensor split feasibility problem.Let C and Q be non-empty closed convex set and A be a semi-symmetric tensor.The tensor split feasibility problem is to find x∈C such that Axm−1∈Q.If we simply take this problem as a special case of the nonlinear split feasibility problem,then we can directly get a projection method to solve it.However,applying this kind of projection method to solve the tensor split feasibility problem is not so efficient.So we propose a Levenberg–Marquardt method to achieve higher efficiency.Theoretical analyses are conducted,and some preliminary numerical results show that the Levenberg–Marquardt method has advantage over the common projection method.
In this paper,a method with parameter is proposed for finding the spectral radius of weakly irreducible nonnegative tensors.What is more,we prove this method has an explicit linear convergence rate for indirectly positive tensors.Interestingly,the algorithm is exactly the NQZ method(proposed by Ng,Qi and Zhou in Finding the largest eigenvalue of a non-negative tensor SIAM J Matrix Anal Appl 31:1090–1099,2009)by taking a specific parameter.Furthermore,we give a modified NQZ method,which has an explicit linear convergence rate for nonnegative tensors and has an error bound for nonnegative tensors with a positive Perron vector.Besides,we promote an inexact power-type algorithm.Finally,some numerical results are reported.
In this paper,we present several sharper upper bounds for the M-spectral radius and Z-spectral radius based on the eigenvalues of some unfolding matrices of nonnegative tensors.Meanwhile,we show that these bounds could be tight for some special tensors.For a general nonnegative tensor which can be transformed into a matrix,we prove the maximal singular value of this matrix is an upper bound of its Z-eigenvalues.Some examples are provided to show these proposed bounds greatly improve some existing ones.
